18,793 research outputs found
Random walk boundaries: their entropies and connections with Hecke pairs
We present three papers in non-singular dynamics concerning boundaries of random walks on locally compact, second countable groups. One common theme is entropy. Paper II and III are concerned with boundary entropy spectra, while Paper I studies topological properties of entropy. In Paper II we moreover establish a technique to relate random walks on locally profinite groups to random walks on dense discrete subgroups, by the concept of Hecke pairs, which is also used in Paper III.In Paper I we introduce different perspectives and extensions of Furstenberg\u27s entropy and show semi-continuity and continuity results in these contexts. In particular we apply these to upper and lower limits of non-nested sequences of sigma-algebras in the sense of Kudo.Paper II relates certain random walks on locally profinite groups to random walks on dense discrete subgroups, using a Hecke subgroup, such that the Poisson boundary of the first becomes a boundary of the second one. If the Poisson boundaries of these two walks happen to coincide, then the Hecke subgroup in charge has to be amenable. For some random walks on lamplighter and solvable Baumslag-Solitar groups we obtain that their Poisson boundary is prime and the quasi-regular representation is reducible. Moreover, we find a group such that for any given summable sequence of positive numbers there is a random walk whose boundary entropy spectrum equals the subsum set of this sequence. In particular we obtain a boundary entropy spectrum which is a Cantor set and one which is an interval.In Paper III we study the boundary entropy spectra of finitely supported, generating random walks on a certain affine group, realizing them as finite subsum sets. We show that the averaged information function of a stationary probability measure does not change when passing to a non-singular, absolutely continuous sigma-finite measure and deduce an entropy formula
Branching Random Walks on Free Products of Groups
We study certain phase transitions of branching random walks (BRW) on Cayley
graphs of free products. The aim of this paper is to compare the size and
structural properties of the trace, i.e., the subgraph that consists of all
edges and vertices that were visited by some particle, with those of the
original Cayley graph. We investigate the phase when the growth parameter
is small enough such that the process survives but the trace is not
the original graph. A first result is that the box-counting dimension of the
boundary of the trace exists, is almost surely constant and equals the
Hausdorff dimension which we denote by . The main result states
that the function has only one point of discontinuity which is
at where is the radius of convergence of the Green function
of the underlying random walk. Furthermore, is bounded by one half
the Hausdorff dimension of the boundary of the original Cayley graph and the
behaviour of as is classified.
In the case of free products of infinite groups the end-boundary can be
decomposed into words of finite and words of infinite length. We prove the
existence of a phase transition such that if
the end boundary of the trace consists only of infinite words and if
it also contains finite words. In the last case,
the Hausdorff dimension of the set of ends (of the trace and the original
graph) induced by finite words is strictly smaller than the one of the ends
induced by infinite words.Comment: 39 pages, 4 figures; final version, accepted for publication in the
Proceedings of LM
Random Walk in an Alcove of an Affine Weyl Group, and Non-Colliding Random Walks on an Interval
We use a reflection argument, introduced by Gessel and Zeilberger, to count
the number of k-step walks between two points which stay within a chamber of a
Weyl group. We apply this technique to walks in the alcoves of the classical
affine Weyl groups. In all cases, we get determinant formulas for the number of
k-step walks. One important example is the region m>x_1>x_2>...>x_n>0, which is
a rescaled alcove of the affine Weyl group C_n. If each coordinate is
considered to be an independent particle, this models n non-colliding random
walks on the interval (0,m). Another case models n non-colliding random walks
on the circle.Comment: v.2, 22 pages; correction in a definition led to changes in many
formulas, also added more background, references, and example
Poisson-Furstenberg boundary and growth of groups
We study the Poisson-Furstenberg boundary of random walks on permutational
wreath products. We give a sufficient condition for a group to admit a
symmetric measure of finite first moment with non-trivial boundary, and show
that this criterion is useful to establish exponential word growth of groups.
We construct groups of exponential growth such that all finitely supported (not
necessarily symmetric, possibly degenerate) random walks on these groups have
trivial boundary. This gives a negative answer to a question of Kaimanovich and
Vershik.Comment: 24 page
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